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SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 22218bh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22218.bh4 | 22218bh1 | \([1, 0, 0, -316882, 111971828]\) | \(-23771111713777/22848457968\) | \(-3382391787572013552\) | \([2]\) | \(506880\) | \(2.2516\) | \(\Gamma_0(N)\)-optimal |
22218.bh3 | 22218bh2 | \([1, 0, 0, -5913702, 5533051680]\) | \(154502321244119857/55101928644\) | \(8157062992429104516\) | \([2, 2]\) | \(1013760\) | \(2.5982\) | |
22218.bh2 | 22218bh3 | \([1, 0, 0, -6765392, 3834952158]\) | \(231331938231569617/90942310746882\) | \(13462725819128930848098\) | \([2]\) | \(2027520\) | \(2.9447\) | |
22218.bh1 | 22218bh4 | \([1, 0, 0, -94611132, 354202649010]\) | \(632678989847546725777/80515134\) | \(11919129439644126\) | \([2]\) | \(2027520\) | \(2.9447\) |
Rank
sage: E.rank()
The elliptic curves in class 22218bh have rank \(1\).
Complex multiplication
The elliptic curves in class 22218bh do not have complex multiplication.Modular form 22218.2.a.bh
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.